On the exponential metric increasing property.
type: Item
instance of: scholarly article
title: On the exponential metric increasing property.
zbMATH Open document ID: 1052.15013
DOI: 10.1016/S0024-3795(03)00647-5
author: Rajendra Bhatia
published in: Linear Algebra and its Applications
publication date: 2003-12-04T00:00:00Z
review text: A short and beautifully simple matrix theoretical proof of the negative curvature of the set of positive definite real or complex matrices \(P\) is given. It is based on the exponential operator of a matrix, the exponential metric increasing property, and the logarithmic mean and geometric-arithmetic mean inequalities for scalars and matrices. The Riemannn metric is studied on the manifold \(P\) of positive definite matrices, as well as a generalized exponential metric increasing property for symmetric gauge functions \(\Phi\). Consequently \(P\) is shown to also be a metric space of non-positive curvature in any Finsler metric \(\delta_\Phi\). The last section derives the Golden-Thompson inequality from these results and investigates related majorization results.
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 53B21
Mathematics Subject Classification ID: 53B40
zbMATH DE Number: 2013054
zbMATH Keywords: exponential metric
zbMATH Keywords: positive definite matrix
zbMATH Keywords: matrix inequality
zbMATH Keywords: geometric mean
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: negative curvature
zbMATH Keywords: matrix norm
zbMATH Keywords: symmetric gauge function
zbMATH Keywords: matrix exponential
zbMATH Keywords: majorization
zbMATH Keywords: exponential map
zbMATH Keywords: unitarily invariant norm
reviewed by: Frank Uhlig
MaRDI profile type: Publication
cites work: Q4169818
cites work: Concavity of certain maps on positive definite matrices and applications to Hadamard products
cites work: Geometrical significance of the Löwner-Heinz inequality
cites work: On an inequality of Lieb and Thirring
cites work: Positive Definite Functions and Operator Inequalities
cites work: Q4720067
cites work: GEODESICS AND OPERATOR MEANS IN THE SPACE OF POSITIVE OPERATORS
cites work: Convexity of the geodesic distance on spaces of positive operators
cites work: Q4352369
cites work: Comparison of various means for operators
cites work: Q4226699
cites work: The Geometric Mean, Matrices, Metrics, and More
cites work: Q4110023
cites work: Q5848223
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description: scientific article; zbMATH DE number 2013054